
where \(\alpha _k(t)\)  and \(\psi _k(x)\)  refer to the \(k\) th modal coefficient and \(k\) th proper orthogonal decomposition (POD) basis function, respectively. Figure REF  shows the the first eight most energetic POD basis functions utilized in this study. We note that, without losing generality, one can also use Fourier harmonics [1]} or randomized orthogonal functions [2]} to forge a set of basis functions.
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